How are electromagnetism and space-time related, and can
electromagnetic fields exert a force on space-time?

A popular science fiction technology is to use electromagnetic fields to
manipulate the local geometry of space-time. All that general relativity
says is that any form of energy can cause space-time to become curved. Too
much curvature and you may end up with black holes or worm holes. The form
of the energy can be anything; raw mass, the energy latent in electromagnetic
fields, neutrinos etc. So, electromagnetic fields are not intimately related
to space-time any more than mass is; its just another form of energy that can
serve to generate curvature via its own gravitational field.